#508883

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,25,20,34,36,21,26,37,42,23,28,38,43)(19,29,32,44,41,40,48,45,31,24,39,30,33,35,46,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,32,25,41,34,48,21,31,37,39,23,33,38,46)(18,24,27,30,20,35,36,47,26,29,42,44,28,40,43,45);(3,7)(6,8)(10,14)(13,16)(17,20)(18,25)(19,30)(21,28)(22,27)(23,26)(24,32)(29,39)(31,44)(33,47)(34,43)(35,46)(36,38)(37,42)(40,48)(41,45);(1,3,4,7)(2,6,5,8)(18,26)(20,28)(24,29)(27,42)(30,44)(35,40)(36,43)(45,47)

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=ei,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416079);